Theorem ltexnq 9797

Description: Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by NM, 24-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ltexnq	|- ( B e. Q. -> ( A <Q B <-> E. x ( A +Q x ) = B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem ltexnq

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 9748	. . . 4 |- <Q C_ ( Q. X. Q. )
2	1	brel 5168	. . 3 |- ( A <Q B -> ( A e. Q. /\ B e. Q. ) )
3		ordpinq 9765	. . . 4 |- ( ( A e. Q. /\ B e. Q. ) -> ( A <Q B <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) )
4		elpqn 9747	. . . . . . . . 9 |- ( A e. Q. -> A e. ( N. X. N. ) )
5	4	adantr 481	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. ) -> A e. ( N. X. N. ) )
6		xp1st 7198	. . . . . . . 8 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
7	5, 6	syl 17	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` A ) e. N. )
8		elpqn 9747	. . . . . . . . 9 |- ( B e. Q. -> B e. ( N. X. N. ) )
9	8	adantl 482	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. ) -> B e. ( N. X. N. ) )
10		xp2nd 7199	. . . . . . . 8 |- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. )
11	9, 10	syl 17	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` B ) e. N. )
12		mulclpi 9715	. . . . . . 7 |- ( ( ( 1st ` A ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 1st ` A ) .N ( 2nd ` B ) ) e. N. )
13	7, 11, 12	syl2anc 693	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( 1st ` A ) .N ( 2nd ` B ) ) e. N. )
14		xp1st 7198	. . . . . . . 8 |- ( B e. ( N. X. N. ) -> ( 1st ` B ) e. N. )
15	9, 14	syl 17	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` B ) e. N. )
16		xp2nd 7199	. . . . . . . 8 |- ( A e. ( N. X. N. ) -> ( 2nd ` A ) e. N. )
17	5, 16	syl 17	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` A ) e. N. )
18		mulclpi 9715	. . . . . . 7 |- ( ( ( 1st ` B ) e. N. /\ ( 2nd ` A ) e. N. ) -> ( ( 1st ` B ) .N ( 2nd ` A ) ) e. N. )
19	15, 17, 18	syl2anc 693	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( 1st ` B ) .N ( 2nd ` A ) ) e. N. )
20		ltexpi 9724	. . . . . 6 |- ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) e. N. /\ ( ( 1st ` B ) .N ( 2nd ` A ) ) e. N. ) -> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) <-> E. y e. N. ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N y ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) ) )
21	13, 19, 20	syl2anc 693	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) <-> E. y e. N. ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N y ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) ) )
22		relxp 5227	. . . . . . . . . . . 12 |- Rel ( N. X. N. )
23	4	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> A e. ( N. X. N. ) )
24		1st2nd 7214	. . . . . . . . . . . 12 |- ( ( Rel ( N. X. N. ) /\ A e. ( N. X. N. ) ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
25	22, 23, 24	sylancr 695	. . . . . . . . . . 11 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
26	25	oveq1d 6665	. . . . . . . . . 10 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( A +pQ <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) = ( <. ( 1st ` A ) , ( 2nd ` A ) >. +pQ <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) )
27	7	adantr 481	. . . . . . . . . . 11 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( 1st ` A ) e. N. )
28	17	adantr 481	. . . . . . . . . . 11 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( 2nd ` A ) e. N. )
29		simpr 477	. . . . . . . . . . 11 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> y e. N. )
30		mulclpi 9715	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` A ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
31	17, 11, 30	syl2anc 693	. . . . . . . . . . . 12 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
32	31	adantr 481	. . . . . . . . . . 11 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
33		addpipq 9759	. . . . . . . . . . 11 |- ( ( ( ( 1st ` A ) e. N. /\ ( 2nd ` A ) e. N. ) /\ ( y e. N. /\ ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. ) ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. +pQ <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) = <. ( ( ( 1st ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) +N ( y .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) >. )
34	27, 28, 29, 32, 33	syl22anc 1327	. . . . . . . . . 10 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. +pQ <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) = <. ( ( ( 1st ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) +N ( y .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) >. )
35	26, 34	eqtrd 2656	. . . . . . . . 9 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( A +pQ <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) = <. ( ( ( 1st ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) +N ( y .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) >. )
36		oveq2 6658	. . . . . . . . . . . 12 |- ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N y ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) -> ( ( 2nd ` A ) .N ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N y ) ) = ( ( 2nd ` A ) .N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) )
37		distrpi 9720	. . . . . . . . . . . . 13 |- ( ( 2nd ` A ) .N ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N y ) ) = ( ( ( 2nd ` A ) .N ( ( 1st ` A ) .N ( 2nd ` B ) ) ) +N ( ( 2nd ` A ) .N y ) )
38		fvex 6201	. . . . . . . . . . . . . . 15 |- ( 2nd ` A ) e. _V
39		fvex 6201	. . . . . . . . . . . . . . 15 |- ( 1st ` A ) e. _V
40		fvex 6201	. . . . . . . . . . . . . . 15 |- ( 2nd ` B ) e. _V
41		mulcompi 9718	. . . . . . . . . . . . . . 15 |- ( x .N y ) = ( y .N x )
42		mulasspi 9719	. . . . . . . . . . . . . . 15 |- ( ( x .N y ) .N z ) = ( x .N ( y .N z ) )
43	38, 39, 40, 41, 42	caov12 6862	. . . . . . . . . . . . . 14 |- ( ( 2nd ` A ) .N ( ( 1st ` A ) .N ( 2nd ` B ) ) ) = ( ( 1st ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) )
44		mulcompi 9718	. . . . . . . . . . . . . 14 |- ( ( 2nd ` A ) .N y ) = ( y .N ( 2nd ` A ) )
45	43, 44	oveq12i 6662	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` A ) .N ( ( 1st ` A ) .N ( 2nd ` B ) ) ) +N ( ( 2nd ` A ) .N y ) ) = ( ( ( 1st ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) +N ( y .N ( 2nd ` A ) ) )
46	37, 45	eqtr2i 2645	. . . . . . . . . . . 12 |- ( ( ( 1st ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) +N ( y .N ( 2nd ` A ) ) ) = ( ( 2nd ` A ) .N ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N y ) )
47		mulasspi 9719	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) = ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( 1st ` B ) ) )
48		mulcompi 9718	. . . . . . . . . . . . . 14 |- ( ( 2nd ` A ) .N ( 1st ` B ) ) = ( ( 1st ` B ) .N ( 2nd ` A ) )
49	48	oveq2i 6661	. . . . . . . . . . . . 13 |- ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( 1st ` B ) ) ) = ( ( 2nd ` A ) .N ( ( 1st ` B ) .N ( 2nd ` A ) ) )
50	47, 49	eqtri 2644	. . . . . . . . . . . 12 |- ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) = ( ( 2nd ` A ) .N ( ( 1st ` B ) .N ( 2nd ` A ) ) )
51	36, 46, 50	3eqtr4g 2681	. . . . . . . . . . 11 |- ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N y ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) -> ( ( ( 1st ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) +N ( y .N ( 2nd ` A ) ) ) = ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) )
52		mulasspi 9719	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) = ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) )
53	52	eqcomi 2631	. . . . . . . . . . . 12 |- ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) = ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) )
54	53	a1i 11	. . . . . . . . . . 11 |- ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N y ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) -> ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) = ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) )
55	51, 54	opeq12d 4410	. . . . . . . . . 10 |- ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N y ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) -> <. ( ( ( 1st ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) +N ( y .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) >. = <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. )
56	55	eqeq2d 2632	. . . . . . . . 9 |- ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N y ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) -> ( ( A +pQ <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) = <. ( ( ( 1st ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) +N ( y .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) >. <-> ( A +pQ <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) = <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. ) )
57	35, 56	syl5ibcom 235	. . . . . . . 8 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N y ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) -> ( A +pQ <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) = <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. ) )
58		fveq2 6191	. . . . . . . . 9 |- ( ( A +pQ <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) = <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. -> ( /Q ` ( A +pQ <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) = ( /Q ` <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. ) )
59		adderpq 9778	. . . . . . . . . . 11 |- ( ( /Q ` A ) +Q ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) = ( /Q ` ( A +pQ <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) )
60		nqerid 9755	. . . . . . . . . . . . 13 |- ( A e. Q. -> ( /Q ` A ) = A )
61	60	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( /Q ` A ) = A )
62	61	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( ( /Q ` A ) +Q ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) = ( A +Q ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) )
63	59, 62	syl5eqr 2670	. . . . . . . . . 10 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( /Q ` ( A +pQ <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) = ( A +Q ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) )
64		mulclpi 9715	. . . . . . . . . . . . . . . 16 |- ( ( ( 2nd ` A ) e. N. /\ ( 2nd ` A ) e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` A ) ) e. N. )
65	17, 17, 64	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( 2nd ` A ) .N ( 2nd ` A ) ) e. N. )
66	65	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` A ) ) e. N. )
67	15	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( 1st ` B ) e. N. )
68	11	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( 2nd ` B ) e. N. )
69		mulcanenq 9782	. . . . . . . . . . . . . 14 |- ( ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) e. N. /\ ( 1st ` B ) e. N. /\ ( 2nd ` B ) e. N. ) -> <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. ~Q <. ( 1st ` B ) , ( 2nd ` B ) >. )
70	66, 67, 68, 69	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. ~Q <. ( 1st ` B ) , ( 2nd ` B ) >. )
71	8	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> B e. ( N. X. N. ) )
72		1st2nd 7214	. . . . . . . . . . . . . 14 |- ( ( Rel ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
73	22, 71, 72	sylancr 695	. . . . . . . . . . . . 13 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
74	70, 73	breqtrrd 4681	. . . . . . . . . . . 12 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. ~Q B )
75		mulclpi 9715	. . . . . . . . . . . . . . 15 |- ( ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) e. N. /\ ( 1st ` B ) e. N. ) -> ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) e. N. )
76	66, 67, 75	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) e. N. )
77		mulclpi 9715	. . . . . . . . . . . . . . 15 |- ( ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) e. N. )
78	66, 68, 77	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) e. N. )
79		opelxpi 5148	. . . . . . . . . . . . . 14 |- ( ( ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) e. N. /\ ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) e. N. ) -> <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. e. ( N. X. N. ) )
80	76, 78, 79	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. e. ( N. X. N. ) )
81		nqereq 9757	. . . . . . . . . . . . 13 |- ( ( <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. ~Q B <-> ( /Q ` <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. ) = ( /Q ` B ) ) )
82	80, 71, 81	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. ~Q B <-> ( /Q ` <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. ) = ( /Q ` B ) ) )
83	74, 82	mpbid 222	. . . . . . . . . . 11 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( /Q ` <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. ) = ( /Q ` B ) )
84		nqerid 9755	. . . . . . . . . . . 12 |- ( B e. Q. -> ( /Q ` B ) = B )
85	84	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( /Q ` B ) = B )
86	83, 85	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( /Q ` <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. ) = B )
87	63, 86	eqeq12d 2637	. . . . . . . . 9 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( ( /Q ` ( A +pQ <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) = ( /Q ` <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. ) <-> ( A +Q ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) = B ) )
88	58, 87	syl5ib 234	. . . . . . . 8 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( ( A +pQ <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) = <. ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 1st ` B ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` A ) ) .N ( 2nd ` B ) ) >. -> ( A +Q ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) = B ) )
89	57, 88	syld 47	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N y ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) -> ( A +Q ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) = B ) )
90		fvex 6201	. . . . . . . 8 |- ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) e. _V
91		oveq2 6658	. . . . . . . . 9 |- ( x = ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) -> ( A +Q x ) = ( A +Q ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) )
92	91	eqeq1d 2624	. . . . . . . 8 |- ( x = ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) -> ( ( A +Q x ) = B <-> ( A +Q ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) = B ) )
93	90, 92	spcev 3300	. . . . . . 7 |- ( ( A +Q ( /Q ` <. y , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. ) ) = B -> E. x ( A +Q x ) = B )
94	89, 93	syl6 35	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. ) /\ y e. N. ) -> ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N y ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) -> E. x ( A +Q x ) = B ) )
95	94	rexlimdva 3031	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> ( E. y e. N. ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N y ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) -> E. x ( A +Q x ) = B ) )
96	21, 95	sylbid 230	. . . 4 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) -> E. x ( A +Q x ) = B ) )
97	3, 96	sylbid 230	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A <Q B -> E. x ( A +Q x ) = B ) )
98	2, 97	mpcom 38	. 2 |- ( A <Q B -> E. x ( A +Q x ) = B )
99		eleq1 2689	. . . . . . 7 |- ( ( A +Q x ) = B -> ( ( A +Q x ) e. Q. <-> B e. Q. ) )
100	99	biimparc 504	. . . . . 6 |- ( ( B e. Q. /\ ( A +Q x ) = B ) -> ( A +Q x ) e. Q. )
101		addnqf 9770	. . . . . . . 8 |- +Q : ( Q. X. Q. ) --> Q.
102	101	fdmi 6052	. . . . . . 7 |- dom +Q = ( Q. X. Q. )
103		0nnq 9746	. . . . . . 7 |- -. (/) e. Q.
104	102, 103	ndmovrcl 6820	. . . . . 6 |- ( ( A +Q x ) e. Q. -> ( A e. Q. /\ x e. Q. ) )
105		ltaddnq 9796	. . . . . 6 |- ( ( A e. Q. /\ x e. Q. ) -> A <Q ( A +Q x ) )
106	100, 104, 105	3syl 18	. . . . 5 |- ( ( B e. Q. /\ ( A +Q x ) = B ) -> A <Q ( A +Q x ) )
107		simpr 477	. . . . 5 |- ( ( B e. Q. /\ ( A +Q x ) = B ) -> ( A +Q x ) = B )
108	106, 107	breqtrd 4679	. . . 4 |- ( ( B e. Q. /\ ( A +Q x ) = B ) -> A <Q B )
109	108	ex 450	. . 3 |- ( B e. Q. -> ( ( A +Q x ) = B -> A <Q B ) )
110	109	exlimdv 1861	. 2 |- ( B e. Q. -> ( E. x ( A +Q x ) = B -> A <Q B ) )
111	98, 110	impbid2 216	1 |- ( B e. Q. -> ( A <Q B <-> E. x ( A +Q x ) = B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   <.cop 4183   class class class wbr 4653   X. cxp 5112   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   N.cnpi 9666   +N cpli 9667   .N cmi 9668   <N clti 9669   +pQ cplpq 9670   ~Q ceq 9673   Q.cnq 9674   /Qcerq 9676   +Q cplq 9677   <Q cltq 9680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-ltnq 9740

This theorem is referenced by:  ltbtwnnq  9800  prnmadd  9819  ltexprlem4  9861  ltexprlem7  9864  prlem936  9869